12
D. Rocchesso: Sound Processing
analog system
signal flowchart
impulse invariance
bilinear transformation
numeric implementations, even local instabilities can be a problem, since the nu-
merical approximations introduced in the representations of variables can easily
produce diverging signals that are difficult to control.
1.5
Continuous-time to discrete-time system con-
version
In many applications, and in particular in sound synthesis by physical modeling,
the design of a discrete-time system starts from the description of a physical
continuous-time system by means of differential equations and constraints. This
description of an analog system can itself be derived from the simplification
of the physical reality into an assembly of basic mechanical elements, such as
springs, dampers, frictions, nonlinearities, etc. . Alternatively, our continuous-
time physical template can result from measurements on a real physical system.
In any case, in order to construct a discrete-time system capable to reproduce
the behavior of the continuous-time physical system, we need to transform the
differential equations into difference equations, in such a way that the resulting
model can be expressed as a signal flowchart in discrete time.
The techniques that are most widely used in signal processing to discretize
a continuous-time LTI system are the impulse invariance and the bilinear trans-
formation.
1.5.1
Impulse Invariance
In the method of the impulse invariance, the impulse response h(n) of the
discrete-time system is a uniform sampling of the impulse response h
s
(t) of
the continuous-time system, rescaled by the width of the sampling interval T ,
according to
h(n) = T h
s
(nT ) .
(30)
In the usual practice of digital filter design, the constant T is usually neglected,
since the design stems from specifications for the discrete-time filter, and the
conversion to continuous time is only an intermediate stage. Since one should
introduce 1/T when going from discrete to continuous time, and T when return-
ing to discrete time, the overall effect of the constant is canceled. Vice versa, if
we start from a description in continuous time, such as in physical modeling,
the constant T should be considered.
From the sampling theorem we can easily deduce that the frequency response
of the discrete-time system is the periodic replication of the frequency response
of the continuous-time system, with a repetition period equal to F
s
= 1/T . In
terms of "discrete-time frequency" (in radians per sample), we can write
H() =
k=-
H
s
j
T
+ j
2
T
k =
k=-
H
s
(j + j2F
s
k) .
(31)
The equation (31) shows that the frequency components in the two domains,
discrete and continuous, can be identical in the base band only if the continuous-
time system is bandlimited. If this is not the case (and it is almost never the
case!), there will be some aliasing that introduces spurious components in the